A010878 a(n) = n mod 9.

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5

Offset

0,3

Comments

Periodic with period of length 9. The digital root of n (A010888) is a very similar sequence.

The rightmost digit in the base-9 representation of n. Also, the equivalent value of the two rightmost digits in the base-3 representation of n. - Hieronymus Fischer, Jun 11 2007

Links

Ely Golden, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Formula

Complex representation: a(n)=(1/9)*(1-r^n)*sum{1<=k<9, k*product{1<=m<9,m<>k, (1-r^(n-m))}} where r=exp(2*pi/9*i) and i=sqrt(-1). Trigonometric representation: a(n)=(256/9)^2*(sin(n*pi/9))^2*sum{1<=k<9, k*product{1<=m<9,m<>k, (sin((n-m)*pi/9))^2}}. G.f.: g(x)=(sum{1<=k<9, k*x^k})/(1-x^9). Also: g(x)=x(8x^9-9x^8+1)/((1-x^9)(1-x)^2). - Hieronymus Fischer, May 31 2007

a(n) = n mod 3 + 3*(floor(n/3)mod 3) = A010872(n) + 3*A010872(A002264(n)). - Hieronymus Fischer, Jun 11 2007

a(n) = floor(12345678/999999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013

a(n) = floor(1513361/96855122*9^(n+1)) mod 9. - Hieronymus Fischer, Jan 04 2013

Maple

A010878 := proc(n)
    modp(n, 9) ;
end proc:
seq(A010878(n), n=0..100) ; # R. J. Mathar, Sep 09 2015

Mathematica

Array[Mod[#, 9]&, 105, 0] (* Jean-François Alcover, Jan 30 2018 *)
PadRight[{}, 120, Range[0, 8]] (* Harvey P. Dale, Dec 19 2018 *)

Prog

(Haskell)
a010878 = (`mod` 9)
a010878_list = cycle [0..8]  -- Reinhard Zumkeller, Jan 09 2013
(PARI) a(n)=n%9 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Partial sums: A130487. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486.

Sequence in context: A207505 A235049 A031087 * A309788 A326746 A257849

Adjacent sequences: A010875 A010876 A010877 * A010879 A010880 A010881

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved